Concept

Self heating — where it appears

The rise in a component's temperature caused by the power it dissipates itself, rather than by its surroundings. It turns a temperature coefficient into an error that depends on the signal, and it arrives with the thermal time constant, so a short pulse can end before it does.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The shunt's resistance as a function of what it is measuring. computed by solving, not by drawing, as a fixed point: the shunt dissipates I²R, its temperature rises by 20 K per watt, and at 50 ppm/K its resistance rises with its temperature — so the resistance the reading is divided by depends on the reading. Iterated to convergence it agrees with the closed form R₀/(1 − αθI²R₀) to 2.2e-16. Along the burden-voltage optimum, where R = u⁄I, the dissipation is I·u rather than I²R, so the temperature rise is 155 mK per ampere and the error is the FIRST power of the current — fitted exponent 1.0007 over five decades. That is the only one of the shunt's errors with the current in it, and it puts a term in I² into the reading, which is a curvature no single-current calibration removes. The upper curve is a shunt of fixed resistance, where the error is quadratic. The fixed point stops existing at 129 kA and never at a current a shunt will see.

The resistance that depends on the reading

Three of a shunt's errors are free of the current being measured, which is the whole content of the burden-voltage optimum. The fourth is not: the shunt dissipates, warms, and its resistance rises — so the divisor the reading uses is a function of the reading. Solved as a fixed point it agrees with R₀/(1 − αθI²R₀) to 2×10⁻¹⁶, and along the optimum, where the dissipation is I·u* rather than I²R, the error is the FIRST power of the current: 7.75 ppm at an ampere, 775 at a hundred, fitted exponent 1.0007.

instruments · Current sensing

A 1 ms pulse of 100 A warms the element 98.7 mK: 4.94 ppm at its end, 2.86 ppm averaged, against 775 ppm held. computed by solving, not by drawing, from the exact step response of the shunt's thermal network — element, terminations and board, 0.2, 4 and 15.8 K/W with 5 mJ/K, 0.1 J/K and 2 J/K — expanded into its Foster modes. A 100 A pulse of 1 ms through the 77.5 µΩ shunt the burden optimum picks, dissipating I·u* = 0.775 W: the element rises 98.7 mK by the pulse's end and cools after it. At 50 ppm/K that is 4.94 ppm of error at the end of the pulse and 2.86 ppm averaged over it, where the same current held long enough for the board to settle would read 775 ppm high.

The pulse that ends before the heat

A shunt's self-heating error is a fixed point in the steady state — 775 ppm at 100 A for the shunt the burden optimum picks, through 20 K/W. A pulse never reaches it. Heat leaves the element through a ladder network — the element's own 5 mJ/K, its terminations, the board — and a pulse short against the element's millisecond heats it adiabatically, by P·t over its heat capacity alone: 0.737 ppm for 100 µs at 100 A, a thousandth of the held figure, 10.7 ppm for 10 ms. Along the burden optimum the error is linear in the current, so the current a pulse may carry for 100 ppm is exact: 12.9 A held, 936 A for 10 ms, 13.6 kA for 100 µs. A reading averaged over the pulse carries half the error of one taken at its end while the heating is adiabatic, and the section-by-section thermal model a data sheet quotes is 10% wrong at 7 ms.

instruments · Current sensing

Named alongside it

The objects these essays reach for when they reach for this one.

Temperature coefficient Thermal resistance Current sensing Current shunt Fixed point Measurement error Model range Nonlinearity Thermal time constant

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